Luke has taught high school algebra and geometry, college calculus, and has a master's degree in education.
You can't always rely on your smartphone to tell you how far you need to travel to get from point A to point B. The distance formula will tell you the distance between any two points on a graph. In this lesson, you'll learn where it comes from and how to use it!
The Pythagorean Theorem and Distance
Now, I like technology as much as the next guy and have been known to use my smartphone to tell me how to get to the grocery store just four blocks away, but even so, every once in a while, I enjoy going camping and getting out of cell phone range for a few days to get away from it all. I actually recently went hiking up and over the Continental Divide close to Boulder, Colorado with a good friend of mine.
We were planning our route for the day, and we picked a spot on our map that looked like a cool place to spend the night. Thing was, it looked kind of far away, and we weren't sure if we'd be able to make it in one day. On top of that, our phones weren't working and couldn't tell us how far away it was. On the map, it was diagonal from us, which meant we couldn't just count over how many boxes over it was.
That meant it was time for some good, old-fashioned Pythagorean Theorem: a^2 + b^2 = c^2. By counting straight over and then straight up, we had a right triangle, and the distance between us and the campsite was just the side length c. A few quick calculations later, and we found that we were about 8.5 miles away - not close, but doable!
Using the Pythagorean Theorem to find distance.
Now, at this point in the lesson, you might be wondering, 'Isn't this supposed to be about the distance formula?' Why are we using the Pythagorean Theorem? Well, the reason I show you this is because the distance formula is really just a condensed version of the Pythagorean Theorem. It takes all the steps we just did and combines them into one formula.
Because it turns so many steps into one line of math, it's actually a pretty messy formula and can be easy to make mistakes on. So, before I show you the formula, I wanted you to see where it comes from. Think of the distance formula as just a shortcut. If you ever forget the shortcut or feel like the shortcut isn't for you, you can always just use the Pythagorean Theorem - like the way I showed you.
The Distance Formula
So here it is, the distance formula:
Now, although it seems like one big mess of letters and math, just think of it as c = sqrt(a^2 + b^2). The way we find a in the distance formula is just doing (x2 - x1), and the way we find b is just (y2 - y1). This is basically what we just did previously by counting on the map.
So, if we were to find the distance to the campsite with the formula this time, it would look like this:
Applying the distance formula
(x2 - x1) would just be (1 - 8), and (y2 - y1) would be (7 - 2). Those two little problems give us -7 and 5, and then squaring both those numbers gives us 49 and 25. Look familiar? Adding those two together gives us 74, and then taking the square root finally leaves us with 8.6, just like before.
The only difference between this way and the Pythagorean Theorem was that we had a -7 for a few steps doing it this way, whereas it was a +7 before. That will happen a lot of the time, but it doesn't matter, because when we multiply it by itself (when we square it), it turns positive anyway; the negative goes away.
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And, that's it! That's the distance formula. Let's just try a few quick examples with the formula to give you a little practice.
You'll usually see the problems stated like this: Find the distance between the points (5, 9) and (-2, 3). Now, we could just graph these two points, go straight over and straight up to make a triangle, and then use a^2 + b^2 to find c^2, like the Pythagorean Theorem, but let's get some practice using the distance formula instead.
We'll start by identifying x1, x2, y1 and y2. x1 is the first x value and x2 the second; same thing for the ys. Now we plug these values into their places in the formula, giving us this: d = sqrt((-2 - 5)^2 + (3 - 9)^2). Doing the subtraction leaves us with -7 and -6. Then, squaring these numbers turns them into 49 and 36. Adding them together makes 85, and then taking the square root gives us our answer as around 9.2.
Not too bad, right? Let's do one more for good measure. Find the distance between (-2, -1) and (2, -4). Again, start by identifying x1, x2, y1 and y2. Then, substitute them into the formula: d = sqrt((2 - (-2))^2 + ((-4) - (-1))^2). This time, we've got to be extra careful with our negatives. 2 - (-2) is like 2 + 2, so that gives us 4, and -4 - (-1) is like -4 + 1, which is -3. That leaves us here, with 4 squared and -3 squared on the inside of the square root. These turn into 16 and 9, which add together to 25, and the square root of 25 is just plain old 5.
And, that's the distance formula! Let's quickly review what we've learned. The distance formula is a condensed version of the Pythagorean Theorem (a^2 + b^2 = c^2) and looks like this: d = sqrt((x2 - x1)^2 + (y2 - y1)^2). x1, x2, y1 and y2 are just the x and y coordinates of these two points. And finally, be careful when you have to plug negatives into the distance formula. Sometimes two negatives make a positive.
At the end of this lesson, you will be able to:
Explain how the distance formula relates to the Pythagorean Theorem
Identify the distance formula
Calculate the distance between points using the distance formula
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